The present invention relates to a method of estimating the configuration of an optical element for use in an optical writing device included in an image forming apparatus, an optical element, and a method of producing the same
Generally, a digital copier, laser printer or similar image forming apparatus includes an optical writing device that includes scanning optics. Today, it is a common practice with the scanning optics to use an aspherical lens represented by:                               x          ⁡                      (            h            )                          =                                            Ch              2                                      1              +                                                1                  -                                                            (                                              1                        +                        k                                            )                                        ⁢                                          C                      2                                        ⁢                                          h                      2                                                                                                    +                      ∑                                          e                i                            ⁢                              h                i                                                                        Eq        .                  xe2x80x83                ⁢                  (          1          )                    
where h denotes a lens height, C denotes a paraxial curvature, k denotes a conic constant, and ei denotes the coefficient of a polynomial.
An aspherical lens is mainly implemented as a plastic lens produced by injection molding.
To meet the increasing demand for high image quality and low cost scanning optics, it is necessary to reduce the number of optical elements constituting the scanning optics. This can be done with, e.g., a toric surface that is aspherical in the main scanning direction, but has a curvature varying in accordance with the main scanning direction. A lens with a toric surface is expressed as:                               z          ⁡                      (                          x              ,              y                        )                          =                                            Cx              2                                      1              +                                                1                  -                                                            (                                              1                        +                        k                                            )                                        ⁢                                          C                      2                                        ⁢                                          x                      2                                                                                                    +                                    ∑                              j                =                0                            m                        ⁢                                          ∑                                  i                  =                  0                                n                            ⁢                                                e                  ji                                ⁢                                  x                  i                                ⁢                                  y                  j                                                                                        Eq        .                  xe2x80x83                ⁢                  (          2          )                    
where x denotes coordinates in the main scanning direction, y denotes coordinates in the subscanning direction, z denotes coordinates in the direction of an optical axis, c denotes a curvature in the main scanning direction, k denotes a conic constant, and eji denotes the coefficient of a polynomial.
An ultraprecision, free curved surface machining apparatus has made it possible to implement the optical function surface of a lens as a toric surface with a high degree of freedom.
Each optical function surface of a lens or optical element included in the scanning optics usually has an effective range extending over several ten millimeters to several hundred millimeters in the main scanning direction, but only over several ten millimeters in the subscanning direction. When such a lens is produced by, e.g., plastic injection molding, a deviation from a designed value (configuration error hereinafter) occurs due to, e.g., the uneven contraction of resin.
Resins in general have a contraction ratio of about 0.7%. To fabricate a specular surface frame for injection molding in accordance with the designed configuration of an optical function surface, it has been customary to implement a configuration enlarged by similar enlargement using a contraction ratio that is determined either by theory or by experience. However, because an optical function surface is as great as several hundred millimeters, the nonlinear contraction of resin introduces a configuration error of several micrometers to several ten of micrometers in the optical function surface, noticeably affecting the performance of the optics. In this respect, the estimation of the configuration error of a lens is essential. It is a common practice to measure the contour configuration of a molding in a plane that passes the design original in the mains scanning direction (generator) by use of a contour measuring device and then estimate a configuration error in accordance with the measured contour configuration. Typical of the contour measuring device is FORM TALYSURF available from Rank Taylor.
Japanese Patent Laid-Open Publication Nos. 6-129944 and 7-35541 each disclose a specific estimation method for determining whether or not a configuration error is sufficiently small. One of conventional estimation methods optimizes the paraxial curvature radius of an aspherical equation in such a manner as to minimize the square sum of a configuration error (best fit R hereinafter). Another conventional method uses as an estimation parameter a difference between the best fit R and an aspherical equation, i.e., a configuration error. However, as the amount of asphyericality, i.e., a deviation from a designed spherical surface increases, the correlation between the belt fit R or the parameter for estimation and the optical performance decreases. Consequently, when a tolerance is distributed in order to prevent performance from being rejected by, e.g., optical simulation, some lenses may be rejected by configuration estimation, but may be allowed as to optical characteristic.
Japanese Patent Laid-Open Publication No. 9-89713, for example, proposes an estimation method using a parameter more closely correlated to optical characteristics than the above-discussed parameters. This estimation method produces the second-order derivative of a configuration error or a quadratic differential based on a difference between adjoining coordinates data or a difference between the derivative and the differential. The method then confines the second-order derivative, the quadratic differential or the difference in a particular range so as to guarantee lens performance. The above document, however, simply teaches that the estimation method implements a closer correlation than the other conventional methods by presenting actual specific moldings. How the above method is effective for lens surfaces other than the actual moldings is not known.
On the other hand, Japanese Patent Laid-Open Publication Nos. 5-96572 and 7-60857 each propose to reduce the configuration error of a molding by measuring the configuration of the molding to thereby determine a configuration error and then correcting a specular surface frame in such a manner as to cancel the configuration error This, however, requires a specular surface frame matching with all optical function surfaces to be corrected until the configuration error becomes sufficiently small.
Japanese Patent Laid-Open Publication Nos. 10-288749 and 11-77842 address to a refractive index distribution particular to a plastic lens produced by injection molding. Specifically, a non-uniform refractive index distribution in resin derives a lens effect and causes the focal point of optics to vary. To solve this problem, it is necessary to revise design or to execute an extra step for the reduction of the refractive index distribution after molding. Even if all optical elements are configured as designed, an optical scanning device including the optical elements cannot achieve expected optical performance unless the refractive index distribution is reduced to zero. This is because the shift of the focal point causes an image surface to curve. Particularly, assume that a spot diameter on an image surface is reduced to implement high density, scanning optics. Then, the F number of the optics must be reduced because the spot diameter is substantially proportional to the F number. This, however, causes the focal depth to decrease in proportion to the square of the spot diameter. Therefore, the prerequisite with the development of such scanning optics is that the curve of the image surface ascribable to the shift of a focal point in the image surface be sufficiently small.
It is therefore a first object of the present invention to provide a configuration estimating method capable of estimating the optical characteristics of a lens on the basis of the result of measurement and estimating the performance of the lens on the basis of the estimated characteristics and therefore adaptive to any desired lens surface and closely correlated to the optical characteristics, and a device for practicing the same.
It is a second object of the present invention to provide a method of producing an optical element capable of sufficiently reducing the curve of an image surface ascribable to the shift of a focal point on the image surface and producing such an optical element in a short period of time at low cost.
In accordance with the present invention, a method of estimating the surface configuration of a lens begins with a step of measuring the contour configuration of the lens. The configuration error of the lens that is a deviation from a designed configuration is determined. Subsequently, a partial curvature at each lens height is determined on the basis of the configuration error. Thereafter, a curvature proportional coefficient, which is a shift of a focal point for a unit curvature at each lens height, as measured in the direction of an optical axis on an image surface, is determined. Finally, the shift is estimated on the basis of the curvature proportional coefficient and partial curvature.